I work in a small group that uses Dropbox for temporary storage and collaboration. Files are constantly being updated. Occasionally I see from my notifications, conflicted copies appearing all over the place (people are saving/editing at the same time).

How can I deal with this (is there some locking/messaging system I can apply) ? I really do not want to be cleaning the conflicted copies up at the close of every business day.

If the group is indeed small, there is a simple workaround that can be helpful (Tiresome for many folks). Since the documents are shared and by default there is no way to know who has a doc open, you can agree to flag the doc by right-clicking and using RENAME... then including your name initials. Example:

Document.doc (original file name)

John Smith is going to edit the file but doesn't want to create a conflict copy. He renames the file to:

JS-Document

Then opens the doc, and works on it. Mary Brown opens the shared Dropbox folder and sees the initials JS, so she understands that John has the doc open and she decides to work on another file in the meantime. After 10 minutes, John's done, he closes the file and renames back to Document.doc... now Mary sees the doc has no initials, so she renames the file to MB-Document, then allowing John and the other team members to notice she's editing the file, and so on. Requires team discipline.

Consider a modern true file versioning system such as git.
There are gui tools available that make such systems much easier to use for non-technical folks, such as gitx (OSX) or gitg (Linux).
Also make sure that folks are not already using any kind of git, svn, etc. system (if they are programmers) as those two approaches, simultaneously will lead to major conflicts.

Folks will need training, the amount will depend on how technical are. As the business grows and people develop both more documents and more technical skills a more formal approach to document changes becomes more important. This is an area where the more formal technical approach already used by programmers can be adapted and adopted by others in the organization.